Shear stress and the deviator

Let the second invariant of the stress deviator be expressed in terms of its principal values, that is, by
$$
\text{\MakeUppercase{\romannumeral 2}}_{\text{S}} = \text{S}_{\text{\MakeUppercase{\romannumeral 1}}}\text{S}_{\text{\MakeUppercase{\romannumeral 2}}} + \text{S}_{\text{\MakeUppercase{\romannumeral 2}}}\text{S}_{\text{\MakeUppercase{\romannumeral 3}}} + \text{S}_{\text{\MakeUppercase{\romannumeral 3}}} \text{S}_{\text{\MakeUppercase{\romannumeral 1}}}.
$$
Show that this sum is the negative of two-thirds the sum of squares of the principal shear stresses.
Is this really true? I used Mathematica to calculate this for an arbitrary symmetric matrix but it didn't turn out true.